Problem: Perform the row operation, $2R_2\rightarrow R_2$, on the following matrix. $\left[\begin{array} {ccc} 2 & 6 & 2 & 1 \\ 7 & 4 & 0 & 0 \\ 2 & 3 & -9 & 1 \end{array} \right] $
Solution: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_2$ is given below. $R_2=\left[\begin{array} {ccc} 7 & 4 & 0 & 0 \end{array} \right]$ We are asked to perform the row operation, $2R_2\rightarrow R_2$. Therefore, we must multiply $R_2$ by $2$. $\begin{aligned}2R_2 &= 2\left[\begin{array} {ccc} 7 & 4 & 0 & 0 \end{array} \right] \\\\&=\left[\begin{array} {ccc} 14 & 8 & 0 & 0 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_2$ with $2R_2$. $\left[\begin{array} {ccc} 2 & 6 & 2 & 1 \\ {7} & {4} & {0} & {0} \\ 2 & 3 & -9 & 1 \end{array} \right] \xrightarrow{2R_2 \rightarrow R_2} \left[\begin{array} {ccc} 2 & 6 & 2 & 1 \\ {14} & {8} & {0} & {0} \\ 2 & 3 & -9 & 1 \end{array} \right] $ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} 2 & 6 & 2 & 1 \\ 14 & 8 & 0 & 0 \\ 2 & 3 & -9 & 1 \end{array} \right]$